Worksheet: The Power of a Point Theorem

In this worksheet, we will practice finding the power of a point with respect to a circle.

Q1:

A point is at a distance 40 from the center of a circle.
If its power with respect to the circle is 81, what is the
radius of the circle, rounded to the nearest integer?

Q2:

A circle with center 𝑀
and a point 𝐴 satisfy 𝑀𝐴=28cm and 𝑃(𝐴)=4.
Using 𝜋=227, find the area and the circumference of the circle to the nearest integer.

AArea =88cm, circumference =176cm

BArea =2,451cm, circumference =176cm

CArea =2,451cm, circumference =88cm

DArea =4,903cm, circumference =88cm

Q3:

A circle has center 𝑀 and radius 𝑟=21.
Find the power of the point 𝐴 with respect to the circle given that 𝐴𝑀=25.

Q4:

A circle with center 𝑁
has a diameter equal to 38 cm.
A point 𝐵 satisfies 𝑁𝐵=7cm.
Find the power of 𝐵 with respect to the circle, giving your answer to the nearest integer.

Q5:

A circle with center 𝑀 has a radius of 8 cm. The power of a point 𝐴 with respect to the circle is 36. Decide whether 𝐴 is outside, inside, or on the circle and then find the distance between 𝐴 and 𝑀.

AOn the circle, 28 cm

BInside the circle, 44 cm

COutside the circle, 10 cm

Q6:

The power of the points 𝐴, 𝐵, and 𝐶
with respect to the circle 𝐾
are 𝑃(𝐴)=4,
𝑃(𝐵)=14, and 𝑃(𝐶)=−1.
The circle 𝐾 has center 𝑀
and a radius of 10 cm.
Calculate the distance between 𝑀 and each of the points.

A𝐴𝑀=2√26cm,
𝐵𝑀=√114cm,
𝐶𝑀=3√11cm

B𝐴𝑀=104cm,
𝐵𝑀=114cm,
𝐶𝑀=99cm

C𝐴𝑀=√14cm,
𝐵𝑀=2√6cm,
𝐶𝑀=3cm

D𝐴𝑀=14cm,
𝐵𝑀=24cm,
𝐶𝑀=9cm

Q7:

Determine the position of a point 𝐴 with respect to the circle 𝑁 if 𝑃(𝐴)=814.

Aoutside the circle

Binside the circle

Con the circumference of the circle

Q8:

The power of a point with respect to a circle is −575 when
its distance from the center of that circle is 84. What is
the circle’s diameter to the nearest hundredth?

Q9:

Given that the point 𝐴 is outside the circle 𝑀, and 𝐴𝐷 is a tangent to the circle at 𝐷 such that 𝐴𝐷=17.65cm,
find the power of the point 𝐴 with respect to the circle 𝑀. Round your answer to the nearest hundredth.

Q10:

A circle with center 𝑀 has a radius of 11 cm.
Point 𝐴 lies 5 cm
from 𝑀 and belongs to the chord 𝐵𝐶. Given that 𝐴𝐵=5𝐴𝐶, calculate 𝐵𝐶, giving your answer to the nearest hundredth.

Q11:

Two circles 𝑀 and 𝑁 intersect at points
𝐴 and 𝐵, and the point 𝐶 satisfies
𝐶∈𝐵𝐴 and 𝐶∉𝐵𝐴.
𝐷 and 𝐸 are the points where
𝐶𝐸 intersects the circle 𝑀 and
𝐶𝐹 is a tangent to 𝑁. Given that
𝐶𝐷=7 and 𝐷𝐸=12, find
𝑃(𝐶).

Q12:

How many circles of radius 5.2 cm are there on points 𝐴, 𝐵 with 𝐴𝐵=24cm?

Q13:

A line 𝐿 intersects a circle with center 𝑀. Point 𝐴 lies
on 𝐿 and is inside the circle. If the radius of the circle is 8 cm,
𝑀𝐴⟂𝐿, and we set 𝑀𝐴=(3𝑥−5)cm,
in what interval does the value of 𝑥 belong?

A−53,133

B53,133

C−53,133

D53,133

Q14:

A circle has a radius of 90 cm.
A point lies on the circle at a distance of (3𝑥−3) cm from the center. Which of the following is true?